Find the modulus of the complex number one plus 𝑖.
Let’s draw our complex plane, where the 𝑥-axis contains all the real numbers, the 𝑦-axis contains all the imaginary numbers, and the other numbers in our plane are complex numbers, which are neither real nor pure imaginary.
We can mark the number that we’re interested in, which is one plus 𝑖. The modulus of this complex number is its distance from the origin of this complex plane, which represents zero. So I’ve drawn in the line segment from zero to one plus 𝑖.
We can draw in a right triangle whose sides both have length one. And so by the Pythagorean theorem, the length of the line segment between zero and one plus 𝑖, which is the distance from zero to one plus 𝑖, is the square root of one squared plus one squared, which is the square root of two. The modulus of the complex number one plus 𝑖 is therefore the square root of two.
In general, the modulus of a complex number 𝑎 plus 𝑏𝑖 is the square root of 𝑎 squared plus 𝑏 squared, and you can check that using this formula does indeed give the same result, the square root of two.
We’re not done with this question yet; we’re also required to find the argument of this complex number. The argument of a complex number is the measure of the angle between that complex number and the positive real axis.
We already have a right triangle drawn with the opposite and adjacent side lengths given; they’re both one. And so the measure 𝜃 of this angle is the inverse tangent function of one over one, which we can find using a calculator, assuming that we’ve got it set in radians, to be 𝜋 by four radians.
In general, the argument of a complex number 𝑎 plus 𝑏𝑖 is the inverse tangent of 𝑏 of 𝑎 if 𝑎 is greater than zero and the inverse tangent of 𝑏 over 𝑎 plus 𝜋 if 𝑎 is less than zero.
The final part of our question is, hence, write the complex number one plus 𝑖 in polar form. The polar form of a complex number 𝑧 is 𝑟 times cos 𝜃 plus 𝑖 sin 𝜃, where 𝑟 is the modulus of the complex number 𝑧 and 𝜃 is its argument. And as we found in the earlier part of the question that the modulus of one plus 𝑖 is the square root of two and the argument of one plus 𝑖 is 𝜋 by four, finding the polar form of one plus 𝑖 is just a case of substituting these values in. So we find that the complex number one plus 𝑖 in polar form is root two times cos 𝜋 by four plus 𝑖 sin 𝜋 by four.
And it’s straightforward to check that this really does represent one plus 𝑖 by expanding out the polar form. You might also like to work out the geometric significance of the real and imaginary parts of this number in polar form, so what is the geometric significance of root two times cos 𝜋 by four and root two times sin 𝜋 by four on the diagram we have involving the complex plane.